If I am divorced with sole legal custody can I move out of state without the permission of my children's father?

It depends on the court order. If the order mandates that the child not leave the state, then the answer is no. However, be aware that if the father was granted rights of visitation, the onus of accommodating that visitation may be on you. If you move far away so as to deny the father his court-ordered visitation, you may be in contempt of the court order. The best bet is to obtain court approval by having the visitation modified.